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Real Numbers - Coggle Diagram

- - - - (a)^2= (3q)^2
        
        a^2= 9q^2
        
        a^2= 3( 3q^2)
        
        a^2= 3m
    - - a= 3q+1
        
        a^2= (3q+1)^2
        
        a^2= 9q^2 + 6q + 1
        
        a^2= 3( 3q^2+2q) + 1
        
        a^2= 3m + 1
- - - - 12 = 2 x 2 x 3
      - Therefore, HCF of 10 and 12 = 2
      - 10 = 2 x 5
- - - - 31/10= 31/2x5
        
        Since the denominator has 2 and 5 as its prime factors, 31/10 has a terminating decimal expansion.
- - - - Let us assume that √3 is rational, Hence √3 = a/b ..................(1)
      - Where a and b are prime integers and no comman factor between a and b.
      - Hence √3b = a
      - By squaring both sides, 3b^2 = a^2 or a^2 / 3 = b^2 ..............(2)
      - equation (2) shows 3 is a factor of a^2 . If a is a prime number, then 3 also should be factor of a.
      - If 3 is a factor of a, then this contradicts the assumption we made for equation (1), that a is a prime number.
      - Hence √3 is not a rational number , it is an irrational number